Integrand size = 26, antiderivative size = 682 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{-\frac {1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {\sqrt [3]{-\frac {1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{8748\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {4+2^{2/3} \sqrt [3]{3} x}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {i \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{4374 \sqrt {3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{8748 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}} \]
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Time = 1.24 (sec) , antiderivative size = 682, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2122, 652, 632, 210, 212} \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{4374 \sqrt {3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {i \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{8748 \sqrt {6} \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}}+\frac {\sqrt [3]{-\frac {1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {\sqrt [3]{-\frac {1}{3}} \left ((-2)^{2/3} \sqrt [3]{3} x+4\right )}{8748\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {2^{2/3} \sqrt [3]{3} x+4}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]
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Rule 210
Rule 212
Rule 632
Rule 652
Rule 2122
Rubi steps \begin{align*} \text {integral}& = 1586874322944 \int \left (-\frac {\sqrt [3]{-\frac {1}{3}} x}{1542441841901568\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2}-\frac {1}{4627325525704704 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )}-\frac {\sqrt [3]{-\frac {1}{3}} x}{1542441841901568\ 2^{2/3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2}-\frac {i}{4627325525704704 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {x}{1542441841901568\ 2^{2/3} \sqrt [3]{3} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2}+\frac {1}{41645929731342336 \sqrt [3]{2} 3^{2/3} \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}\right ) \, dx \\ & = -\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {x}{\left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3}}+\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{26244 \sqrt [3]{2} 3^{2/3}}+\frac {\int \frac {x}{\left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )^2} \, dx}{8748\ 2^{2/3} \sqrt [3]{3}}-\frac {i \int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{2916 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\sqrt [3]{-\frac {1}{3}} \int \frac {x}{\left (-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2\right )^2} \, dx}{972\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac {\int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{2916 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4} \\ & = \frac {\sqrt [3]{-\frac {1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {\sqrt [3]{-\frac {1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{17496\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {4+2^{2/3} \sqrt [3]{3} x}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {\text {Subst}\left (\int \frac {1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right )}{13122 \sqrt [3]{2} 3^{2/3}}+\frac {i \text {Subst}\left (\int \frac {1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{1458 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\text {Subst}\left (\int \frac {1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{1458 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac {\int \frac {1}{6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2} \, dx}{17496 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}-\frac {\int \frac {1}{6+3\ 2^{2/3} \sqrt [3]{3} x+x^2} \, dx}{17496 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}+\frac {\int \frac {1}{-6+3 \sqrt [3]{-3} 2^{2/3} x-x^2} \, dx}{8748 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )} \\ & = \frac {\sqrt [3]{-\frac {1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {\sqrt [3]{-\frac {1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{17496\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {4+2^{2/3} \sqrt [3]{3} x}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {i \tan ^{-1}\left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {\text {Subst}\left (\int \frac {1}{-6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )-x^2} \, dx,x,3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{8748 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}+\frac {\text {Subst}\left (\int \frac {1}{-6 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )-x^2} \, dx,x,3\ 2^{2/3} \sqrt [3]{3}+2 x\right )}{8748 \left (4-3 \sqrt [3]{2} 3^{2/3}\right )}-\frac {\text {Subst}\left (\int \frac {1}{-6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )-x^2} \, dx,x,3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{4374 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )} \\ & = \frac {\sqrt [3]{-\frac {1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {\sqrt [3]{-\frac {1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{17496\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {4+2^{2/3} \sqrt [3]{3} x}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\tan ^{-1}\left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\tan ^{-1}\left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{8748 \sqrt {6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {i \tan ^{-1}\left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{8748 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.24 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {972-144 x+648 x^2+729 x^3-27 x^4+4 x^5}{615276 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}+\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {144 \log (x-\text {$\#$1})-324 \log (x-\text {$\#$1}) \text {$\#$1}+2043 \log (x-\text {$\#$1}) \text {$\#$1}^2-54 \log (x-\text {$\#$1}) \text {$\#$1}^3+4 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{3691656} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.18
method | result | size |
default | \(\frac {\frac {1}{153819} x^{5}-\frac {1}{22788} x^{4}+\frac {1}{844} x^{3}+\frac {2}{1899} x^{2}-\frac {4}{17091} x +\frac {1}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-54 \textit {\_R}^{3}+2043 \textit {\_R}^{2}-324 \textit {\_R} +144\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{3691656}\) | \(122\) |
risch | \(\frac {\frac {1}{153819} x^{5}-\frac {1}{22788} x^{4}+\frac {1}{844} x^{3}+\frac {2}{1899} x^{2}-\frac {4}{17091} x +\frac {1}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-54 \textit {\_R}^{3}+2043 \textit {\_R}^{2}-324 \textit {\_R} +144\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{3691656}\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (463) = 926\).
Time = 0.96 (sec) , antiderivative size = 1445, normalized size of antiderivative = 2.12 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \]
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Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.15 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (27493895104978847349012449000830556700672 t^{6} - 1318718189226950088862983192576 t^{4} + 12120917704776776448 t^{2} - 39753025, \left ( t \mapsto t \log {\left (\frac {947842259001288723909832054550209950242045952 t^{5}}{61864539719962655} - \frac {243458646817775607639654889480814592 t^{4}}{9811980923071} - \frac {41682556475067500431787310779667456 t^{3}}{61864539719962655} + \frac {12026877442664328616462272 t^{2}}{9811980923071} + \frac {216142618488859793668428 t}{61864539719962655} + x - \frac {308574300024117}{39247923692284} \right )} \right )\right )} + \frac {4 x^{5} - 27 x^{4} + 729 x^{3} + 648 x^{2} - 144 x + 972}{615276 x^{6} + 11074968 x^{4} + 199349424 x^{3} + 66449808 x^{2} + 132899616} \]
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\[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{5}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
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\[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{5}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
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Time = 0.19 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.44 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\left (\sum _{k=1}^6\ln \left (-\frac {4477969\,\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}{189282278088}+\frac {6305\,x}{4967524106141472}-\frac {\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )\,x\,16340881}{5110621508376}-\frac {{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^2\,x\,43348696}{10818603}-\frac {{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^3\,x\,65333687616}{44521}-\frac {{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^4\,x\,40024496812032}{211}-{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^5\,x\,6940988288557056+\frac {5943884\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^2}{400689}+\frac {224442467136\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^3}{44521}-\frac {137087493272064\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^4}{211}-168897381688221696\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^5-\frac {13082875}{178830867821092992}\right )\,\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )\right )+\frac {\frac {x^5}{153819}-\frac {x^4}{22788}+\frac {x^3}{844}+\frac {2\,x^2}{1899}-\frac {4\,x}{17091}+\frac {1}{633}}{x^6+18\,x^4+324\,x^3+108\,x^2+216} \]
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